\(\int \frac {(a+b x^2)^{3/2}}{(c+d x^2)^{5/2} (e+f x^2)^{3/2}} \, dx\) [518]

Optimal result

Integrand size = 34, antiderivative size = 521 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^{5/2} \left (e+f x^2\right )^{3/2}} \, dx=-\frac {(b c-a d) x \sqrt {a+b x^2}}{3 c (d e-c f) \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}+\frac {2 (a d (d e-3 c f)+b c (d e+c f)) x \sqrt {a+b x^2}}{3 c^2 (d e-c f)^2 \sqrt {c+d x^2} \sqrt {e+f x^2}}+\frac {\sqrt {-b e+a f} \left (2 b c e (d e+3 c f)+a \left (2 d^2 e^2-7 c d e f-3 c^2 f^2\right )\right ) \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} E\left (\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right )|\frac {a (d e-c f)}{c (b e-a f)}\right )}{3 \sqrt {a} c e (d e-c f)^3 \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}}-\frac {\sqrt {-b e+a f} (a d (d e-9 c f)+2 b c (d e+3 c f)) \sqrt {a+b x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-b e+a f} x}{\sqrt {a} \sqrt {e+f x^2}}\right ),\frac {a (d e-c f)}{c (b e-a f)}\right )}{3 \sqrt {a} c (d e-c f)^3 \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}} \] Output:

-1/3*(-a*d+b*c)*x*(b*x^2+a)^(1/2)/c/(-c*f+d*e)/(d*x^2+c)^(3/2)/(f*x^2+e)^( 
1/2)+2/3*(a*d*(-3*c*f+d*e)+b*c*(c*f+d*e))*x*(b*x^2+a)^(1/2)/c^2/(-c*f+d*e) 
^2/(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2)+1/3*(a*f-b*e)^(1/2)*(2*b*c*e*(3*c*f+d*e 
)+a*(-3*c^2*f^2-7*c*d*e*f+2*d^2*e^2))*(b*x^2+a)^(1/2)*(e*(d*x^2+c)/c/(f*x^ 
2+e))^(1/2)*EllipticE((a*f-b*e)^(1/2)*x/a^(1/2)/(f*x^2+e)^(1/2),(a*(-c*f+d 
*e)/c/(-a*f+b*e))^(1/2))/a^(1/2)/c/e/(-c*f+d*e)^3/(d*x^2+c)^(1/2)/(e*(b*x^ 
2+a)/a/(f*x^2+e))^(1/2)-1/3*(a*f-b*e)^(1/2)*(a*d*(-9*c*f+d*e)+2*b*c*(3*c*f 
+d*e))*(b*x^2+a)^(1/2)*(e*(d*x^2+c)/c/(f*x^2+e))^(1/2)*EllipticF((a*f-b*e) 
^(1/2)*x/a^(1/2)/(f*x^2+e)^(1/2),(a*(-c*f+d*e)/c/(-a*f+b*e))^(1/2))/a^(1/2 
)/c/(-c*f+d*e)^3/(d*x^2+c)^(1/2)/(e*(b*x^2+a)/a/(f*x^2+e))^(1/2)
 

Mathematica [F]

\[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^{5/2} \left (e+f x^2\right )^{3/2}} \, dx=\int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^{5/2} \left (e+f x^2\right )^{3/2}} \, dx \] Input:

Integrate[(a + b*x^2)^(3/2)/((c + d*x^2)^(5/2)*(e + f*x^2)^(3/2)),x]
 

Output:

Integrate[(a + b*x^2)^(3/2)/((c + d*x^2)^(5/2)*(e + f*x^2)^(3/2)), x]
 

Rubi [F]

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^{5/2} \left (e+f x^2\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 434

\(\displaystyle \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^{5/2} \left (e+f x^2\right )^{3/2}}dx\)

Input:

Int[(a + b*x^2)^(3/2)/((c + d*x^2)^(5/2)*(e + f*x^2)^(3/2)),x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 434

Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( 
x_)^2)^(r_.), x_Symbol] :> Unintegrable[(a + b*x^2)^p*(c + d*x^2)^q*(e + f* 
x^2)^r, x] /; FreeQ[{a, b, c, d, e, f, p, q, r}, x]
 

Maple [F]

\[\int \frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{\left (x^{2} d +c \right )^{\frac {5}{2}} \left (f \,x^{2}+e \right )^{\frac {3}{2}}}d x\]

Input:

int((b*x^2+a)^(3/2)/(d*x^2+c)^(5/2)/(f*x^2+e)^(3/2),x)
 

Output:

int((b*x^2+a)^(3/2)/(d*x^2+c)^(5/2)/(f*x^2+e)^(3/2),x)
 

Fricas [F]

\[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^{5/2} \left (e+f x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}}}{{\left (d x^{2} + c\right )}^{\frac {5}{2}} {\left (f x^{2} + e\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((b*x^2+a)^(3/2)/(d*x^2+c)^(5/2)/(f*x^2+e)^(3/2),x, algorithm="fr 
icas")
 

Output:

integral((b*x^2 + a)^(3/2)*sqrt(d*x^2 + c)*sqrt(f*x^2 + e)/(d^3*f^2*x^10 + 
 (2*d^3*e*f + 3*c*d^2*f^2)*x^8 + (d^3*e^2 + 6*c*d^2*e*f + 3*c^2*d*f^2)*x^6 
 + c^3*e^2 + (3*c*d^2*e^2 + 6*c^2*d*e*f + c^3*f^2)*x^4 + (3*c^2*d*e^2 + 2* 
c^3*e*f)*x^2), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^{5/2} \left (e+f x^2\right )^{3/2}} \, dx=\text {Timed out} \] Input:

integrate((b*x**2+a)**(3/2)/(d*x**2+c)**(5/2)/(f*x**2+e)**(3/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^{5/2} \left (e+f x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}}}{{\left (d x^{2} + c\right )}^{\frac {5}{2}} {\left (f x^{2} + e\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((b*x^2+a)^(3/2)/(d*x^2+c)^(5/2)/(f*x^2+e)^(3/2),x, algorithm="ma 
xima")
 

Output:

integrate((b*x^2 + a)^(3/2)/((d*x^2 + c)^(5/2)*(f*x^2 + e)^(3/2)), x)
 

Giac [F]

\[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^{5/2} \left (e+f x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}}}{{\left (d x^{2} + c\right )}^{\frac {5}{2}} {\left (f x^{2} + e\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((b*x^2+a)^(3/2)/(d*x^2+c)^(5/2)/(f*x^2+e)^(3/2),x, algorithm="gi 
ac")
 

Output:

integrate((b*x^2 + a)^(3/2)/((d*x^2 + c)^(5/2)*(f*x^2 + e)^(3/2)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^{5/2} \left (e+f x^2\right )^{3/2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{3/2}}{{\left (d\,x^2+c\right )}^{5/2}\,{\left (f\,x^2+e\right )}^{3/2}} \,d x \] Input:

int((a + b*x^2)^(3/2)/((c + d*x^2)^(5/2)*(e + f*x^2)^(3/2)),x)
 

Output:

int((a + b*x^2)^(3/2)/((c + d*x^2)^(5/2)*(e + f*x^2)^(3/2)), x)
 

Reduce [F]

\[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^{5/2} \left (e+f x^2\right )^{3/2}} \, dx=\int \frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{\left (d \,x^{2}+c \right )^{\frac {5}{2}} \left (f \,x^{2}+e \right )^{\frac {3}{2}}}d x \] Input:

int((b*x^2+a)^(3/2)/(d*x^2+c)^(5/2)/(f*x^2+e)^(3/2),x)
 

Output:

int((b*x^2+a)^(3/2)/(d*x^2+c)^(5/2)/(f*x^2+e)^(3/2),x)